# Equation of a Line in Three Dimensions

Equation of a line is defined as y= mx+c, where c is the y-intercept and m is the slope. Vectors can be defined as a quantity possessing both direction and magnitude. Position vectors simply denote the position or location of a point in the three-dimensional Cartesian system with respect to a reference origin.

It is known that we can uniquely determine a line if:

• It passes through a particular point in a specific direction, or
• It passes through two unique points

Let us study each case separately and try to determine the equation of a line in both the given cases.

Equation of a Line passing through a given point and parallel to a given vector

Let us consider that the position vector of the given point be  $$\vec{a}$$with respect to the origin. The line passing through point A is given by l and it is parallel to the vector $$\vec{k}$$ as shown below. Let us choose any random point R on the line l and its position vector with respect to origin of the rectangular co-ordinate system is given by $$\vec{r}$$.

Since the line segment, $$\overline{AR}$$ is parallel to vector  $$\vec{k}$$, therefore for any real number α,

$$\overline{AR}$$ = α $$\vec{k}$$

Also, $$\overline{AR}$$=$$\overline{OR}$$ – $$\overline{OA}$$

Therefore, α $$\vec{r}$$ = $$\vec{r}$$ – $$\vec{a}$$

From the above equation it can be seen that for different values of α, the above equations give the position of any arbitrary point R lying on the line passing through point A and parallel to vector k. Therefore, the vector equation of a line passing through a given point and parallel to a given vector is given by:

$$\vec{r}$$ = $$\vec{a}$$ + α$$\vec{k}$$

If the three dimensional coordinates of the point A are given as (x1, y1, z1) and the direction cosines of this point is given as a, b, c then considering the rectangular coordinates of point R as (x, y, z)

Substituting these values in the vector equation of a line passing through a given point and parallel to a given vector and equating the coefficients of unit vectors i, j and k, we have,

Eliminating α we have:

This gives us the Cartesian equation of line.

Equation of a Line passing through two given points

Let us consider that the position vector of the two given points A and B be $$\vec{a}$$ and $$\vec{b}$$ with respect to the origin. Let us choose any random point R on the line and its position vector with respect to origin of the rectangular co-ordinate system is given by $$\vec{r}$$ .

Point R lies on the line AB if and only if the vectors  $$\overline{AR}$$ and $$\overline{AB}$$ are collinear. Also,

$$\overline{AR}$$ = $$\vec{r}$$ – $$\vec{a}$$

$$\overline{AB}$$ = $$\vec{b}$$ – $$\vec{a}$$

Thus R lies on AB only if;

$$\vec{r}$$ – $$\vec{a}$$ = α $$\vec{b}$$ – $$\vec{a}$$

Here α is any real number.

From the above equation it can be seen that for different values of α, the above equation gives the position of any arbitrary point R lying on the line passing through point A and B. Therefore, the vector equation of a line passing through two given points is given by:

$$\vec{r}$$ = $$\vec{a} + α ($$\vec{b} \) – $$\vec{a}$$)

If the three dimensional coordinates of the points A and B are given as (x1, y1, z1) and (x2, y2, z2) then considering the rectangular coordinates of point R as (x, y, z)

Substituting these values in the vector equation of a line passing through two given points and equating the coefficients of unit vectors i, j and k, we have

Eliminating α we have:

This gives us the Cartesian equation of a line.